A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
48 ( , ) 1
1 6.2
2636 K R Lr
L R
(3.21)
provided rain rate R10mm/h.
This model has been verified for the calculation of rain attenuation at different percentages of time at different locations referred to by Lin [Lin, 1977] as “city A”, “city B”, etc.
A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
49
( %)A p kR LKit rc (3.23)
where the parameters kand are the power law coefficients, Lis the path length, and Krcthe reduction coefficient that depends on the spatial structure of rain and integration time.
Garcia-Lopez and Peiro [1983] gave their reductionKrcto be:
1
rc
it
K a L bR cL d e
(3.24)
with parameters , , ,a b c dand edepending on the rain rate integration time. The coefficients of these parameters also depend on the geographical area since the spatial structure of rain can be different in different geographical areas [Rogers, 1981]. This makes this prediction model to be applicable to most sites around the globe
Garcia-Lopez and Peiro [1983] obtained the coefficients for equation (3.24) for Europe with an integration time rain rateit1, and is given as:
rc 0.98
40.2 it 13 200 28200
K L R L
(3.25)
This has been tested over 21 European radio links reported by the CCIR in Report 338 and it has been observed to predict the attenuation fairly well. This model has also been tested in the USA and Japan, but with different path reduction coefficients, since these coefficients are geographically dependent [Garcia-Lopez and Peiro, 1983].
Garcia-Lopez et al. [1988], provided two sets of coefficients for , , ,a b c dande. One set for worldwide use (which is adopted by the temperate climates) and the other for Australia. The coefficients for Australia is also adopted for tropical climatic condition (a0.72,b7.6,c 4.75,d2408,and e10,000) [Garcia-Lopez, 1989] for the prediction of rain attenuation.
A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
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3 .5 Rain Attenuation Research in South America
In Brazil, de Miranda et al. [1998b] presented the statistical modeling of the cumulative probability distribution function of rain rate in various sites in Brazil. The sites used in their work provide a wide view of rainfall behaviour in Brazil. The sites are located in the cities of are Belem, Manaus, Recife, Rio de Janeiro, and Sao Paulo. The location of the sites span over 22 degrees of latitude and represented two types of equatorial and three types of tropical climates.
Their results indicate that the ITU-R expectations for the rainfall rate distribution in climates P and N tend to overestimate the measured values for the sites studied [de Miranda et al. 1998b].
Migliora et al. [1990] also confirmed eleven different rain climatic zones in Brazil as against the two climatic zones P and N allocated by the ITU-R. Mello Silva et al. [1990] gave the conversion factors for converting 5-minutes rain rate to a 1-minute integration time rain rate at various probability levels for 5 locations in Brazil.
Assis and Dias [1998] presented a modified ITU-R rain attenuation model for low latitude areas, for terrestrial paths, based on the fact that rain attenuation prediction model currently adopted by ITU-R appears to be inadequate for most tropical regions [Green, 2004]. Assis [1992] suggested that the empirical expression used for scaling the rain rate exceeded for 0.01% of an average year (R0.01) to other percentages of time may cause an overestimation of the predicted rain attenuation in the range from 0.01 to 0.001%. Therefore, Assis and Dias [1998] proposed a modified scaling expression to solve this problem based on 15 years experimental data from Brazil and India. In 2002, da Costa and Assis presented an empirical solution based on a given rain cell model to develop a path length factor. This is a very important parameter when deriving a rain attenuation model due to the non-uniformity of rain along propagation paths.
3.5.1 CETUC Rain Attenuation Model
Perez Garcia and da Silva Mello [2004] developed an attenuation model for terrestrial radio links using data from measurements in tropical climates, in Brazil, together with data from temperate climates. This model aimed at improving the ITU-R and other rain attenuation models developed in the temperate climates. The CETUC rain attenuation model uses complete point rainfall rate cumulative probability distribution as input data to calculate the attenuation distribution, in an equiprobability basis [Perez Garcia and da Silva Mello, 2004]. This model is developed with the assumption that the in-homogeneity of rain along a propagation path length can be modeled by an
A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
51
equivalent uniform rain cell with a rain rate that is dependent on the length of the terrestrial link.
The attenuation model is given as:
r s .
A r L (3.26)
whereAr is the rain attenuation along a terrestrial link in dB, s is the specific attenuation in (dB/km), ris the reduction factor, Lis the actual path length link (km).
The reduction factor is given as;
0.164 ( 0.369 0.115 )
3.445 p L
r L R (3.27)
where Rpis the rainfall rate at the percentage of time of interest, p[Perez Garcia and da Silva Mello 2004].
3.5.2 New CETUC Rain Attenuation Model
In 2007, da Silva Mello et al. proposed an improved semi-empirical method for the prediction of rain attenuation along terrestrial line-of-sight links. This method uses full rain rate distribution from 74 year data set from 64 links at 34 sites in 15 countries [da Silva Mello et al., 2007] to predict the attenuation distribution. This model tends to avoid extrapolations functions which are dependent on the percentage of time of interest as used in the previous CETUC model and the ITU-R model. This model also assumes an equivalent cell of uniform rainfall rate (which is the basis of the ITU-R method [Ajayi et al., 1996; Cost 255, 2002]) and actual length d0 to model for the non-uniformity of rain along propagation path length. Fig. 3-1 below shows the description of an equivalent rain cell. The effective path length deff is given as the average length L of the intersection between the cell and the path.
Therefore, the effective path length is given by:
0 0 0
1 1
( ) 1
d eff
d
d L L x dx r d d
d d d d
(3.28)A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
52
d L x
d0
Fig. 3- 1: Equivalent rain cell [da Silva Mello et al., 2007]
The effective path length deff is always smaller than the actual path lengthd0, as seen from equation (3.28), which thus leads to path reduction factor rto be given as:
0
deff
r d (3.29)
The cumulative distribution of rain attenuation is obtained from the distribution of rain rate along a terrestrial link by:
,
1 0
p eff eff p
p
A d k R R d d
d d R
(2.30)
where:
is the specific attenuation (dB/km),
Reffis the effective rain rate, as a function of Rpand d , Rp is the point rainfall rate exceeded at %p of time, Apis the rain attenuation exceeded at %p of time
kand are the specific attenuation parameters given by ITU-R [ITU-R 838, 2005 ] The empirical expression for the effective rainfall rate is represented by:
A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
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0.753 0.197
1.763 d
Reff R (3.31)
and the equivalent cell diameter d0 is given by a power law expression;
0.244
0 119
d R (3.32)
Using equations (3.31) and (3.32), the rain attenuation distribution along terrestrial paths can be predicted by [da Silva Mello et al., 2007]:
0.753 0.197
0.244
( ) 1.763
1 119
d p
p
A P k R d
d R
(3.33)